Properties

Label 4014.2.a.r.1.5
Level 4014
Weight 2
Character 4014.1
Self dual yes
Analytic conductor 32.052
Analytic rank 1
Dimension 5
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 4014 = 2 \cdot 3^{2} \cdot 223 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4014.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(32.0519513713\)
Analytic rank: \(1\)
Dimension: \(5\)
Coefficient field: 5.5.356173.1
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1338)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(2.23241\)
Character \(\chi\) = 4014.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} +3.02013 q^{5} -3.77138 q^{7} +1.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} +3.02013 q^{5} -3.77138 q^{7} +1.00000 q^{8} +3.02013 q^{10} -1.20245 q^{11} -1.04630 q^{13} -3.77138 q^{14} +1.00000 q^{16} -3.48116 q^{17} -4.38932 q^{19} +3.02013 q^{20} -1.20245 q^{22} -6.93812 q^{23} +4.12120 q^{25} -1.04630 q^{26} -3.77138 q^{28} +1.44714 q^{29} -3.28522 q^{31} +1.00000 q^{32} -3.48116 q^{34} -11.3901 q^{35} -5.93312 q^{37} -4.38932 q^{38} +3.02013 q^{40} -5.58831 q^{41} +9.61326 q^{43} -1.20245 q^{44} -6.93812 q^{46} -1.67635 q^{47} +7.22334 q^{49} +4.12120 q^{50} -1.04630 q^{52} -7.89183 q^{53} -3.63156 q^{55} -3.77138 q^{56} +1.44714 q^{58} -13.9770 q^{59} +10.0775 q^{61} -3.28522 q^{62} +1.00000 q^{64} -3.15995 q^{65} +7.69317 q^{67} -3.48116 q^{68} -11.3901 q^{70} -2.66924 q^{71} +3.48767 q^{73} -5.93312 q^{74} -4.38932 q^{76} +4.53491 q^{77} -7.37800 q^{79} +3.02013 q^{80} -5.58831 q^{82} +2.11619 q^{83} -10.5136 q^{85} +9.61326 q^{86} -1.20245 q^{88} +6.47710 q^{89} +3.94598 q^{91} -6.93812 q^{92} -1.67635 q^{94} -13.2563 q^{95} +8.21487 q^{97} +7.22334 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5q + 5q^{2} + 5q^{4} - 5q^{5} - q^{7} + 5q^{8} + O(q^{10}) \) \( 5q + 5q^{2} + 5q^{4} - 5q^{5} - q^{7} + 5q^{8} - 5q^{10} - 9q^{11} - q^{14} + 5q^{16} - 6q^{17} - 4q^{19} - 5q^{20} - 9q^{22} - 16q^{23} + 8q^{25} - q^{28} - 8q^{29} - q^{31} + 5q^{32} - 6q^{34} - 22q^{35} - 2q^{37} - 4q^{38} - 5q^{40} - 4q^{41} + 3q^{43} - 9q^{44} - 16q^{46} - 18q^{47} + 2q^{49} + 8q^{50} - 26q^{53} + q^{55} - q^{56} - 8q^{58} - 21q^{59} - 20q^{61} - q^{62} + 5q^{64} + 3q^{65} - 5q^{67} - 6q^{68} - 22q^{70} - 17q^{71} + 5q^{73} - 2q^{74} - 4q^{76} - 2q^{77} - 21q^{79} - 5q^{80} - 4q^{82} - 11q^{83} - 12q^{85} + 3q^{86} - 9q^{88} + 5q^{89} - 10q^{91} - 16q^{92} - 18q^{94} - 10q^{95} - 11q^{97} + 2q^{98} + O(q^{100}) \)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) 3.02013 1.35064 0.675322 0.737523i \(-0.264005\pi\)
0.675322 + 0.737523i \(0.264005\pi\)
\(6\) 0 0
\(7\) −3.77138 −1.42545 −0.712725 0.701444i \(-0.752539\pi\)
−0.712725 + 0.701444i \(0.752539\pi\)
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) 3.02013 0.955050
\(11\) −1.20245 −0.362553 −0.181276 0.983432i \(-0.558023\pi\)
−0.181276 + 0.983432i \(0.558023\pi\)
\(12\) 0 0
\(13\) −1.04630 −0.290190 −0.145095 0.989418i \(-0.546349\pi\)
−0.145095 + 0.989418i \(0.546349\pi\)
\(14\) −3.77138 −1.00794
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −3.48116 −0.844305 −0.422153 0.906525i \(-0.638725\pi\)
−0.422153 + 0.906525i \(0.638725\pi\)
\(18\) 0 0
\(19\) −4.38932 −1.00698 −0.503490 0.864001i \(-0.667951\pi\)
−0.503490 + 0.864001i \(0.667951\pi\)
\(20\) 3.02013 0.675322
\(21\) 0 0
\(22\) −1.20245 −0.256364
\(23\) −6.93812 −1.44670 −0.723349 0.690482i \(-0.757398\pi\)
−0.723349 + 0.690482i \(0.757398\pi\)
\(24\) 0 0
\(25\) 4.12120 0.824240
\(26\) −1.04630 −0.205196
\(27\) 0 0
\(28\) −3.77138 −0.712725
\(29\) 1.44714 0.268727 0.134363 0.990932i \(-0.457101\pi\)
0.134363 + 0.990932i \(0.457101\pi\)
\(30\) 0 0
\(31\) −3.28522 −0.590042 −0.295021 0.955491i \(-0.595327\pi\)
−0.295021 + 0.955491i \(0.595327\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) −3.48116 −0.597014
\(35\) −11.3901 −1.92528
\(36\) 0 0
\(37\) −5.93312 −0.975398 −0.487699 0.873012i \(-0.662164\pi\)
−0.487699 + 0.873012i \(0.662164\pi\)
\(38\) −4.38932 −0.712042
\(39\) 0 0
\(40\) 3.02013 0.477525
\(41\) −5.58831 −0.872747 −0.436374 0.899766i \(-0.643737\pi\)
−0.436374 + 0.899766i \(0.643737\pi\)
\(42\) 0 0
\(43\) 9.61326 1.46601 0.733004 0.680224i \(-0.238118\pi\)
0.733004 + 0.680224i \(0.238118\pi\)
\(44\) −1.20245 −0.181276
\(45\) 0 0
\(46\) −6.93812 −1.02297
\(47\) −1.67635 −0.244520 −0.122260 0.992498i \(-0.539014\pi\)
−0.122260 + 0.992498i \(0.539014\pi\)
\(48\) 0 0
\(49\) 7.22334 1.03191
\(50\) 4.12120 0.582826
\(51\) 0 0
\(52\) −1.04630 −0.145095
\(53\) −7.89183 −1.08403 −0.542013 0.840370i \(-0.682338\pi\)
−0.542013 + 0.840370i \(0.682338\pi\)
\(54\) 0 0
\(55\) −3.63156 −0.489680
\(56\) −3.77138 −0.503972
\(57\) 0 0
\(58\) 1.44714 0.190018
\(59\) −13.9770 −1.81966 −0.909828 0.414986i \(-0.863786\pi\)
−0.909828 + 0.414986i \(0.863786\pi\)
\(60\) 0 0
\(61\) 10.0775 1.29029 0.645145 0.764060i \(-0.276797\pi\)
0.645145 + 0.764060i \(0.276797\pi\)
\(62\) −3.28522 −0.417223
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −3.15995 −0.391944
\(66\) 0 0
\(67\) 7.69317 0.939871 0.469935 0.882701i \(-0.344277\pi\)
0.469935 + 0.882701i \(0.344277\pi\)
\(68\) −3.48116 −0.422153
\(69\) 0 0
\(70\) −11.3901 −1.36138
\(71\) −2.66924 −0.316781 −0.158390 0.987377i \(-0.550630\pi\)
−0.158390 + 0.987377i \(0.550630\pi\)
\(72\) 0 0
\(73\) 3.48767 0.408201 0.204100 0.978950i \(-0.434573\pi\)
0.204100 + 0.978950i \(0.434573\pi\)
\(74\) −5.93312 −0.689711
\(75\) 0 0
\(76\) −4.38932 −0.503490
\(77\) 4.53491 0.516801
\(78\) 0 0
\(79\) −7.37800 −0.830089 −0.415045 0.909801i \(-0.636234\pi\)
−0.415045 + 0.909801i \(0.636234\pi\)
\(80\) 3.02013 0.337661
\(81\) 0 0
\(82\) −5.58831 −0.617125
\(83\) 2.11619 0.232282 0.116141 0.993233i \(-0.462948\pi\)
0.116141 + 0.993233i \(0.462948\pi\)
\(84\) 0 0
\(85\) −10.5136 −1.14036
\(86\) 9.61326 1.03662
\(87\) 0 0
\(88\) −1.20245 −0.128182
\(89\) 6.47710 0.686571 0.343285 0.939231i \(-0.388460\pi\)
0.343285 + 0.939231i \(0.388460\pi\)
\(90\) 0 0
\(91\) 3.94598 0.413652
\(92\) −6.93812 −0.723349
\(93\) 0 0
\(94\) −1.67635 −0.172902
\(95\) −13.2563 −1.36007
\(96\) 0 0
\(97\) 8.21487 0.834093 0.417047 0.908885i \(-0.363065\pi\)
0.417047 + 0.908885i \(0.363065\pi\)
\(98\) 7.22334 0.729668
\(99\) 0 0
\(100\) 4.12120 0.412120
\(101\) 1.83630 0.182719 0.0913594 0.995818i \(-0.470879\pi\)
0.0913594 + 0.995818i \(0.470879\pi\)
\(102\) 0 0
\(103\) 1.86823 0.184082 0.0920410 0.995755i \(-0.470661\pi\)
0.0920410 + 0.995755i \(0.470661\pi\)
\(104\) −1.04630 −0.102598
\(105\) 0 0
\(106\) −7.89183 −0.766522
\(107\) 0.540324 0.0522351 0.0261176 0.999659i \(-0.491686\pi\)
0.0261176 + 0.999659i \(0.491686\pi\)
\(108\) 0 0
\(109\) −2.22693 −0.213301 −0.106650 0.994297i \(-0.534013\pi\)
−0.106650 + 0.994297i \(0.534013\pi\)
\(110\) −3.63156 −0.346256
\(111\) 0 0
\(112\) −3.77138 −0.356362
\(113\) 11.0410 1.03865 0.519326 0.854576i \(-0.326183\pi\)
0.519326 + 0.854576i \(0.326183\pi\)
\(114\) 0 0
\(115\) −20.9541 −1.95398
\(116\) 1.44714 0.134363
\(117\) 0 0
\(118\) −13.9770 −1.28669
\(119\) 13.1288 1.20351
\(120\) 0 0
\(121\) −9.55411 −0.868555
\(122\) 10.0775 0.912373
\(123\) 0 0
\(124\) −3.28522 −0.295021
\(125\) −2.65409 −0.237389
\(126\) 0 0
\(127\) −18.1860 −1.61375 −0.806873 0.590725i \(-0.798842\pi\)
−0.806873 + 0.590725i \(0.798842\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) −3.15995 −0.277146
\(131\) −13.8555 −1.21056 −0.605280 0.796012i \(-0.706939\pi\)
−0.605280 + 0.796012i \(0.706939\pi\)
\(132\) 0 0
\(133\) 16.5538 1.43540
\(134\) 7.69317 0.664589
\(135\) 0 0
\(136\) −3.48116 −0.298507
\(137\) −13.6133 −1.16306 −0.581530 0.813525i \(-0.697546\pi\)
−0.581530 + 0.813525i \(0.697546\pi\)
\(138\) 0 0
\(139\) 11.0528 0.937488 0.468744 0.883334i \(-0.344707\pi\)
0.468744 + 0.883334i \(0.344707\pi\)
\(140\) −11.3901 −0.962638
\(141\) 0 0
\(142\) −2.66924 −0.223998
\(143\) 1.25812 0.105209
\(144\) 0 0
\(145\) 4.37054 0.362954
\(146\) 3.48767 0.288641
\(147\) 0 0
\(148\) −5.93312 −0.487699
\(149\) 13.1615 1.07823 0.539115 0.842232i \(-0.318759\pi\)
0.539115 + 0.842232i \(0.318759\pi\)
\(150\) 0 0
\(151\) 10.5025 0.854683 0.427342 0.904090i \(-0.359450\pi\)
0.427342 + 0.904090i \(0.359450\pi\)
\(152\) −4.38932 −0.356021
\(153\) 0 0
\(154\) 4.53491 0.365433
\(155\) −9.92179 −0.796937
\(156\) 0 0
\(157\) 17.9254 1.43060 0.715301 0.698816i \(-0.246289\pi\)
0.715301 + 0.698816i \(0.246289\pi\)
\(158\) −7.37800 −0.586962
\(159\) 0 0
\(160\) 3.02013 0.238762
\(161\) 26.1663 2.06220
\(162\) 0 0
\(163\) 10.0729 0.788973 0.394487 0.918902i \(-0.370922\pi\)
0.394487 + 0.918902i \(0.370922\pi\)
\(164\) −5.58831 −0.436374
\(165\) 0 0
\(166\) 2.11619 0.164248
\(167\) 8.97670 0.694638 0.347319 0.937747i \(-0.387092\pi\)
0.347319 + 0.937747i \(0.387092\pi\)
\(168\) 0 0
\(169\) −11.9053 −0.915790
\(170\) −10.5136 −0.806354
\(171\) 0 0
\(172\) 9.61326 0.733004
\(173\) −22.5083 −1.71127 −0.855637 0.517577i \(-0.826834\pi\)
−0.855637 + 0.517577i \(0.826834\pi\)
\(174\) 0 0
\(175\) −15.5426 −1.17491
\(176\) −1.20245 −0.0906382
\(177\) 0 0
\(178\) 6.47710 0.485479
\(179\) 18.4992 1.38270 0.691348 0.722522i \(-0.257017\pi\)
0.691348 + 0.722522i \(0.257017\pi\)
\(180\) 0 0
\(181\) 19.2505 1.43088 0.715438 0.698676i \(-0.246227\pi\)
0.715438 + 0.698676i \(0.246227\pi\)
\(182\) 3.94598 0.292496
\(183\) 0 0
\(184\) −6.93812 −0.511485
\(185\) −17.9188 −1.31742
\(186\) 0 0
\(187\) 4.18593 0.306105
\(188\) −1.67635 −0.122260
\(189\) 0 0
\(190\) −13.2563 −0.961716
\(191\) −13.2214 −0.956666 −0.478333 0.878179i \(-0.658759\pi\)
−0.478333 + 0.878179i \(0.658759\pi\)
\(192\) 0 0
\(193\) 12.7010 0.914240 0.457120 0.889405i \(-0.348881\pi\)
0.457120 + 0.889405i \(0.348881\pi\)
\(194\) 8.21487 0.589793
\(195\) 0 0
\(196\) 7.22334 0.515953
\(197\) 6.80378 0.484749 0.242375 0.970183i \(-0.422074\pi\)
0.242375 + 0.970183i \(0.422074\pi\)
\(198\) 0 0
\(199\) −18.3936 −1.30389 −0.651944 0.758267i \(-0.726046\pi\)
−0.651944 + 0.758267i \(0.726046\pi\)
\(200\) 4.12120 0.291413
\(201\) 0 0
\(202\) 1.83630 0.129202
\(203\) −5.45771 −0.383056
\(204\) 0 0
\(205\) −16.8774 −1.17877
\(206\) 1.86823 0.130166
\(207\) 0 0
\(208\) −1.04630 −0.0725476
\(209\) 5.27795 0.365084
\(210\) 0 0
\(211\) −23.9426 −1.64827 −0.824137 0.566391i \(-0.808339\pi\)
−0.824137 + 0.566391i \(0.808339\pi\)
\(212\) −7.89183 −0.542013
\(213\) 0 0
\(214\) 0.540324 0.0369358
\(215\) 29.0333 1.98006
\(216\) 0 0
\(217\) 12.3898 0.841075
\(218\) −2.22693 −0.150827
\(219\) 0 0
\(220\) −3.63156 −0.244840
\(221\) 3.64232 0.245009
\(222\) 0 0
\(223\) −1.00000 −0.0669650
\(224\) −3.77138 −0.251986
\(225\) 0 0
\(226\) 11.0410 0.734438
\(227\) −18.9100 −1.25510 −0.627549 0.778577i \(-0.715942\pi\)
−0.627549 + 0.778577i \(0.715942\pi\)
\(228\) 0 0
\(229\) −9.36817 −0.619066 −0.309533 0.950889i \(-0.600173\pi\)
−0.309533 + 0.950889i \(0.600173\pi\)
\(230\) −20.9541 −1.38167
\(231\) 0 0
\(232\) 1.44714 0.0950092
\(233\) 22.4238 1.46903 0.734517 0.678590i \(-0.237409\pi\)
0.734517 + 0.678590i \(0.237409\pi\)
\(234\) 0 0
\(235\) −5.06279 −0.330260
\(236\) −13.9770 −0.909828
\(237\) 0 0
\(238\) 13.1288 0.851013
\(239\) −12.7281 −0.823313 −0.411657 0.911339i \(-0.635050\pi\)
−0.411657 + 0.911339i \(0.635050\pi\)
\(240\) 0 0
\(241\) −14.7087 −0.947472 −0.473736 0.880667i \(-0.657095\pi\)
−0.473736 + 0.880667i \(0.657095\pi\)
\(242\) −9.55411 −0.614161
\(243\) 0 0
\(244\) 10.0775 0.645145
\(245\) 21.8154 1.39374
\(246\) 0 0
\(247\) 4.59253 0.292216
\(248\) −3.28522 −0.208611
\(249\) 0 0
\(250\) −2.65409 −0.167860
\(251\) 14.4741 0.913597 0.456799 0.889570i \(-0.348996\pi\)
0.456799 + 0.889570i \(0.348996\pi\)
\(252\) 0 0
\(253\) 8.34276 0.524505
\(254\) −18.1860 −1.14109
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 4.56495 0.284754 0.142377 0.989813i \(-0.454526\pi\)
0.142377 + 0.989813i \(0.454526\pi\)
\(258\) 0 0
\(259\) 22.3761 1.39038
\(260\) −3.15995 −0.195972
\(261\) 0 0
\(262\) −13.8555 −0.855996
\(263\) −27.4213 −1.69087 −0.845434 0.534080i \(-0.820658\pi\)
−0.845434 + 0.534080i \(0.820658\pi\)
\(264\) 0 0
\(265\) −23.8344 −1.46413
\(266\) 16.5538 1.01498
\(267\) 0 0
\(268\) 7.69317 0.469935
\(269\) −3.32320 −0.202619 −0.101309 0.994855i \(-0.532303\pi\)
−0.101309 + 0.994855i \(0.532303\pi\)
\(270\) 0 0
\(271\) 7.89850 0.479800 0.239900 0.970798i \(-0.422885\pi\)
0.239900 + 0.970798i \(0.422885\pi\)
\(272\) −3.48116 −0.211076
\(273\) 0 0
\(274\) −13.6133 −0.822407
\(275\) −4.95555 −0.298831
\(276\) 0 0
\(277\) −23.2990 −1.39990 −0.699952 0.714190i \(-0.746795\pi\)
−0.699952 + 0.714190i \(0.746795\pi\)
\(278\) 11.0528 0.662904
\(279\) 0 0
\(280\) −11.3901 −0.680688
\(281\) 6.05838 0.361413 0.180706 0.983537i \(-0.442162\pi\)
0.180706 + 0.983537i \(0.442162\pi\)
\(282\) 0 0
\(283\) 20.8294 1.23818 0.619088 0.785321i \(-0.287502\pi\)
0.619088 + 0.785321i \(0.287502\pi\)
\(284\) −2.66924 −0.158390
\(285\) 0 0
\(286\) 1.25812 0.0743942
\(287\) 21.0757 1.24406
\(288\) 0 0
\(289\) −4.88152 −0.287148
\(290\) 4.37054 0.256647
\(291\) 0 0
\(292\) 3.48767 0.204100
\(293\) 12.3205 0.719771 0.359886 0.932996i \(-0.382816\pi\)
0.359886 + 0.932996i \(0.382816\pi\)
\(294\) 0 0
\(295\) −42.2125 −2.45771
\(296\) −5.93312 −0.344855
\(297\) 0 0
\(298\) 13.1615 0.762423
\(299\) 7.25933 0.419818
\(300\) 0 0
\(301\) −36.2553 −2.08972
\(302\) 10.5025 0.604352
\(303\) 0 0
\(304\) −4.38932 −0.251745
\(305\) 30.4354 1.74272
\(306\) 0 0
\(307\) −4.95093 −0.282565 −0.141282 0.989969i \(-0.545123\pi\)
−0.141282 + 0.989969i \(0.545123\pi\)
\(308\) 4.53491 0.258400
\(309\) 0 0
\(310\) −9.92179 −0.563520
\(311\) −14.6555 −0.831037 −0.415518 0.909585i \(-0.636400\pi\)
−0.415518 + 0.909585i \(0.636400\pi\)
\(312\) 0 0
\(313\) 27.3865 1.54798 0.773988 0.633200i \(-0.218259\pi\)
0.773988 + 0.633200i \(0.218259\pi\)
\(314\) 17.9254 1.01159
\(315\) 0 0
\(316\) −7.37800 −0.415045
\(317\) 17.2101 0.966614 0.483307 0.875451i \(-0.339436\pi\)
0.483307 + 0.875451i \(0.339436\pi\)
\(318\) 0 0
\(319\) −1.74011 −0.0974276
\(320\) 3.02013 0.168831
\(321\) 0 0
\(322\) 26.1663 1.45819
\(323\) 15.2799 0.850199
\(324\) 0 0
\(325\) −4.31200 −0.239186
\(326\) 10.0729 0.557888
\(327\) 0 0
\(328\) −5.58831 −0.308563
\(329\) 6.32215 0.348552
\(330\) 0 0
\(331\) −7.49987 −0.412230 −0.206115 0.978528i \(-0.566082\pi\)
−0.206115 + 0.978528i \(0.566082\pi\)
\(332\) 2.11619 0.116141
\(333\) 0 0
\(334\) 8.97670 0.491183
\(335\) 23.2344 1.26943
\(336\) 0 0
\(337\) 2.60559 0.141935 0.0709677 0.997479i \(-0.477391\pi\)
0.0709677 + 0.997479i \(0.477391\pi\)
\(338\) −11.9053 −0.647561
\(339\) 0 0
\(340\) −10.5136 −0.570178
\(341\) 3.95031 0.213922
\(342\) 0 0
\(343\) −0.842304 −0.0454801
\(344\) 9.61326 0.518312
\(345\) 0 0
\(346\) −22.5083 −1.21005
\(347\) 6.30819 0.338641 0.169321 0.985561i \(-0.445843\pi\)
0.169321 + 0.985561i \(0.445843\pi\)
\(348\) 0 0
\(349\) 7.33681 0.392730 0.196365 0.980531i \(-0.437086\pi\)
0.196365 + 0.980531i \(0.437086\pi\)
\(350\) −15.5426 −0.830789
\(351\) 0 0
\(352\) −1.20245 −0.0640909
\(353\) −24.8373 −1.32195 −0.660977 0.750406i \(-0.729858\pi\)
−0.660977 + 0.750406i \(0.729858\pi\)
\(354\) 0 0
\(355\) −8.06147 −0.427858
\(356\) 6.47710 0.343285
\(357\) 0 0
\(358\) 18.4992 0.977714
\(359\) 3.12865 0.165124 0.0825620 0.996586i \(-0.473690\pi\)
0.0825620 + 0.996586i \(0.473690\pi\)
\(360\) 0 0
\(361\) 0.266169 0.0140089
\(362\) 19.2505 1.01178
\(363\) 0 0
\(364\) 3.94598 0.206826
\(365\) 10.5332 0.551334
\(366\) 0 0
\(367\) 10.0031 0.522155 0.261078 0.965318i \(-0.415922\pi\)
0.261078 + 0.965318i \(0.415922\pi\)
\(368\) −6.93812 −0.361675
\(369\) 0 0
\(370\) −17.9188 −0.931554
\(371\) 29.7631 1.54522
\(372\) 0 0
\(373\) 18.1472 0.939628 0.469814 0.882765i \(-0.344321\pi\)
0.469814 + 0.882765i \(0.344321\pi\)
\(374\) 4.18593 0.216449
\(375\) 0 0
\(376\) −1.67635 −0.0864510
\(377\) −1.51413 −0.0779818
\(378\) 0 0
\(379\) −7.77655 −0.399455 −0.199727 0.979852i \(-0.564006\pi\)
−0.199727 + 0.979852i \(0.564006\pi\)
\(380\) −13.2563 −0.680036
\(381\) 0 0
\(382\) −13.2214 −0.676465
\(383\) −29.0604 −1.48492 −0.742458 0.669893i \(-0.766340\pi\)
−0.742458 + 0.669893i \(0.766340\pi\)
\(384\) 0 0
\(385\) 13.6960 0.698014
\(386\) 12.7010 0.646466
\(387\) 0 0
\(388\) 8.21487 0.417047
\(389\) 9.50596 0.481972 0.240986 0.970529i \(-0.422529\pi\)
0.240986 + 0.970529i \(0.422529\pi\)
\(390\) 0 0
\(391\) 24.1527 1.22146
\(392\) 7.22334 0.364834
\(393\) 0 0
\(394\) 6.80378 0.342769
\(395\) −22.2825 −1.12116
\(396\) 0 0
\(397\) 21.6620 1.08718 0.543592 0.839350i \(-0.317064\pi\)
0.543592 + 0.839350i \(0.317064\pi\)
\(398\) −18.3936 −0.921988
\(399\) 0 0
\(400\) 4.12120 0.206060
\(401\) −20.1045 −1.00397 −0.501985 0.864876i \(-0.667397\pi\)
−0.501985 + 0.864876i \(0.667397\pi\)
\(402\) 0 0
\(403\) 3.43731 0.171225
\(404\) 1.83630 0.0913594
\(405\) 0 0
\(406\) −5.45771 −0.270862
\(407\) 7.13429 0.353634
\(408\) 0 0
\(409\) −36.8815 −1.82367 −0.911836 0.410555i \(-0.865335\pi\)
−0.911836 + 0.410555i \(0.865335\pi\)
\(410\) −16.8774 −0.833517
\(411\) 0 0
\(412\) 1.86823 0.0920410
\(413\) 52.7128 2.59383
\(414\) 0 0
\(415\) 6.39118 0.313731
\(416\) −1.04630 −0.0512989
\(417\) 0 0
\(418\) 5.27795 0.258153
\(419\) −33.9553 −1.65883 −0.829413 0.558636i \(-0.811325\pi\)
−0.829413 + 0.558636i \(0.811325\pi\)
\(420\) 0 0
\(421\) −36.8261 −1.79479 −0.897397 0.441225i \(-0.854544\pi\)
−0.897397 + 0.441225i \(0.854544\pi\)
\(422\) −23.9426 −1.16551
\(423\) 0 0
\(424\) −7.89183 −0.383261
\(425\) −14.3466 −0.695910
\(426\) 0 0
\(427\) −38.0061 −1.83924
\(428\) 0.540324 0.0261176
\(429\) 0 0
\(430\) 29.0333 1.40011
\(431\) 1.58581 0.0763860 0.0381930 0.999270i \(-0.487840\pi\)
0.0381930 + 0.999270i \(0.487840\pi\)
\(432\) 0 0
\(433\) −2.99805 −0.144077 −0.0720386 0.997402i \(-0.522950\pi\)
−0.0720386 + 0.997402i \(0.522950\pi\)
\(434\) 12.3898 0.594730
\(435\) 0 0
\(436\) −2.22693 −0.106650
\(437\) 30.4537 1.45680
\(438\) 0 0
\(439\) 17.2951 0.825448 0.412724 0.910856i \(-0.364577\pi\)
0.412724 + 0.910856i \(0.364577\pi\)
\(440\) −3.63156 −0.173128
\(441\) 0 0
\(442\) 3.64232 0.173248
\(443\) −22.8314 −1.08475 −0.542377 0.840135i \(-0.682476\pi\)
−0.542377 + 0.840135i \(0.682476\pi\)
\(444\) 0 0
\(445\) 19.5617 0.927313
\(446\) −1.00000 −0.0473514
\(447\) 0 0
\(448\) −3.77138 −0.178181
\(449\) −28.4260 −1.34150 −0.670752 0.741682i \(-0.734028\pi\)
−0.670752 + 0.741682i \(0.734028\pi\)
\(450\) 0 0
\(451\) 6.71967 0.316417
\(452\) 11.0410 0.519326
\(453\) 0 0
\(454\) −18.9100 −0.887489
\(455\) 11.9174 0.558696
\(456\) 0 0
\(457\) −4.12755 −0.193079 −0.0965393 0.995329i \(-0.530777\pi\)
−0.0965393 + 0.995329i \(0.530777\pi\)
\(458\) −9.36817 −0.437746
\(459\) 0 0
\(460\) −20.9541 −0.976988
\(461\) 23.7885 1.10794 0.553970 0.832537i \(-0.313112\pi\)
0.553970 + 0.832537i \(0.313112\pi\)
\(462\) 0 0
\(463\) −5.25289 −0.244123 −0.122061 0.992523i \(-0.538950\pi\)
−0.122061 + 0.992523i \(0.538950\pi\)
\(464\) 1.44714 0.0671816
\(465\) 0 0
\(466\) 22.4238 1.03876
\(467\) −28.3361 −1.31124 −0.655620 0.755091i \(-0.727592\pi\)
−0.655620 + 0.755091i \(0.727592\pi\)
\(468\) 0 0
\(469\) −29.0139 −1.33974
\(470\) −5.06279 −0.233529
\(471\) 0 0
\(472\) −13.9770 −0.643345
\(473\) −11.5595 −0.531506
\(474\) 0 0
\(475\) −18.0893 −0.829993
\(476\) 13.1288 0.601757
\(477\) 0 0
\(478\) −12.7281 −0.582170
\(479\) −12.1657 −0.555865 −0.277932 0.960601i \(-0.589649\pi\)
−0.277932 + 0.960601i \(0.589649\pi\)
\(480\) 0 0
\(481\) 6.20780 0.283051
\(482\) −14.7087 −0.669964
\(483\) 0 0
\(484\) −9.55411 −0.434278
\(485\) 24.8100 1.12656
\(486\) 0 0
\(487\) −18.2502 −0.826994 −0.413497 0.910505i \(-0.635693\pi\)
−0.413497 + 0.910505i \(0.635693\pi\)
\(488\) 10.0775 0.456187
\(489\) 0 0
\(490\) 21.8154 0.985521
\(491\) 5.52096 0.249157 0.124579 0.992210i \(-0.460242\pi\)
0.124579 + 0.992210i \(0.460242\pi\)
\(492\) 0 0
\(493\) −5.03771 −0.226887
\(494\) 4.59253 0.206628
\(495\) 0 0
\(496\) −3.28522 −0.147511
\(497\) 10.0667 0.451555
\(498\) 0 0
\(499\) 32.2553 1.44394 0.721972 0.691922i \(-0.243236\pi\)
0.721972 + 0.691922i \(0.243236\pi\)
\(500\) −2.65409 −0.118695
\(501\) 0 0
\(502\) 14.4741 0.646011
\(503\) 12.8314 0.572125 0.286063 0.958211i \(-0.407653\pi\)
0.286063 + 0.958211i \(0.407653\pi\)
\(504\) 0 0
\(505\) 5.54587 0.246788
\(506\) 8.34276 0.370881
\(507\) 0 0
\(508\) −18.1860 −0.806873
\(509\) −4.24927 −0.188345 −0.0941727 0.995556i \(-0.530021\pi\)
−0.0941727 + 0.995556i \(0.530021\pi\)
\(510\) 0 0
\(511\) −13.1533 −0.581869
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) 4.56495 0.201351
\(515\) 5.64230 0.248629
\(516\) 0 0
\(517\) 2.01573 0.0886516
\(518\) 22.3761 0.983148
\(519\) 0 0
\(520\) −3.15995 −0.138573
\(521\) 26.2874 1.15167 0.575836 0.817565i \(-0.304677\pi\)
0.575836 + 0.817565i \(0.304677\pi\)
\(522\) 0 0
\(523\) 7.47399 0.326815 0.163407 0.986559i \(-0.447752\pi\)
0.163407 + 0.986559i \(0.447752\pi\)
\(524\) −13.8555 −0.605280
\(525\) 0 0
\(526\) −27.4213 −1.19562
\(527\) 11.4364 0.498176
\(528\) 0 0
\(529\) 25.1376 1.09294
\(530\) −23.8344 −1.03530
\(531\) 0 0
\(532\) 16.5538 0.717700
\(533\) 5.84703 0.253263
\(534\) 0 0
\(535\) 1.63185 0.0705511
\(536\) 7.69317 0.332294
\(537\) 0 0
\(538\) −3.32320 −0.143273
\(539\) −8.68572 −0.374120
\(540\) 0 0
\(541\) −14.4483 −0.621181 −0.310590 0.950544i \(-0.600527\pi\)
−0.310590 + 0.950544i \(0.600527\pi\)
\(542\) 7.89850 0.339269
\(543\) 0 0
\(544\) −3.48116 −0.149254
\(545\) −6.72562 −0.288094
\(546\) 0 0
\(547\) 11.1792 0.477986 0.238993 0.971021i \(-0.423183\pi\)
0.238993 + 0.971021i \(0.423183\pi\)
\(548\) −13.6133 −0.581530
\(549\) 0 0
\(550\) −4.95555 −0.211305
\(551\) −6.35195 −0.270602
\(552\) 0 0
\(553\) 27.8253 1.18325
\(554\) −23.2990 −0.989881
\(555\) 0 0
\(556\) 11.0528 0.468744
\(557\) −7.03110 −0.297917 −0.148959 0.988843i \(-0.547592\pi\)
−0.148959 + 0.988843i \(0.547592\pi\)
\(558\) 0 0
\(559\) −10.0583 −0.425421
\(560\) −11.3901 −0.481319
\(561\) 0 0
\(562\) 6.05838 0.255557
\(563\) 25.3616 1.06886 0.534432 0.845212i \(-0.320526\pi\)
0.534432 + 0.845212i \(0.320526\pi\)
\(564\) 0 0
\(565\) 33.3453 1.40285
\(566\) 20.8294 0.875523
\(567\) 0 0
\(568\) −2.66924 −0.111999
\(569\) −29.5466 −1.23866 −0.619329 0.785131i \(-0.712595\pi\)
−0.619329 + 0.785131i \(0.712595\pi\)
\(570\) 0 0
\(571\) 29.4674 1.23317 0.616586 0.787288i \(-0.288515\pi\)
0.616586 + 0.787288i \(0.288515\pi\)
\(572\) 1.25812 0.0526047
\(573\) 0 0
\(574\) 21.0757 0.879681
\(575\) −28.5934 −1.19243
\(576\) 0 0
\(577\) −0.367741 −0.0153093 −0.00765464 0.999971i \(-0.502437\pi\)
−0.00765464 + 0.999971i \(0.502437\pi\)
\(578\) −4.88152 −0.203045
\(579\) 0 0
\(580\) 4.37054 0.181477
\(581\) −7.98097 −0.331107
\(582\) 0 0
\(583\) 9.48954 0.393017
\(584\) 3.48767 0.144321
\(585\) 0 0
\(586\) 12.3205 0.508955
\(587\) −31.4369 −1.29754 −0.648770 0.760985i \(-0.724716\pi\)
−0.648770 + 0.760985i \(0.724716\pi\)
\(588\) 0 0
\(589\) 14.4199 0.594161
\(590\) −42.2125 −1.73786
\(591\) 0 0
\(592\) −5.93312 −0.243850
\(593\) 39.8376 1.63593 0.817966 0.575266i \(-0.195102\pi\)
0.817966 + 0.575266i \(0.195102\pi\)
\(594\) 0 0
\(595\) 39.6507 1.62552
\(596\) 13.1615 0.539115
\(597\) 0 0
\(598\) 7.25933 0.296856
\(599\) 16.6473 0.680191 0.340096 0.940391i \(-0.389541\pi\)
0.340096 + 0.940391i \(0.389541\pi\)
\(600\) 0 0
\(601\) −41.2749 −1.68364 −0.841818 0.539761i \(-0.818515\pi\)
−0.841818 + 0.539761i \(0.818515\pi\)
\(602\) −36.2553 −1.47766
\(603\) 0 0
\(604\) 10.5025 0.427342
\(605\) −28.8547 −1.17311
\(606\) 0 0
\(607\) −46.5136 −1.88793 −0.943965 0.330045i \(-0.892936\pi\)
−0.943965 + 0.330045i \(0.892936\pi\)
\(608\) −4.38932 −0.178011
\(609\) 0 0
\(610\) 30.4354 1.23229
\(611\) 1.75396 0.0709575
\(612\) 0 0
\(613\) 15.8555 0.640398 0.320199 0.947350i \(-0.396250\pi\)
0.320199 + 0.947350i \(0.396250\pi\)
\(614\) −4.95093 −0.199803
\(615\) 0 0
\(616\) 4.53491 0.182717
\(617\) 16.3571 0.658513 0.329256 0.944241i \(-0.393202\pi\)
0.329256 + 0.944241i \(0.393202\pi\)
\(618\) 0 0
\(619\) 7.77228 0.312394 0.156197 0.987726i \(-0.450077\pi\)
0.156197 + 0.987726i \(0.450077\pi\)
\(620\) −9.92179 −0.398469
\(621\) 0 0
\(622\) −14.6555 −0.587632
\(623\) −24.4276 −0.978672
\(624\) 0 0
\(625\) −28.6217 −1.14487
\(626\) 27.3865 1.09458
\(627\) 0 0
\(628\) 17.9254 0.715301
\(629\) 20.6541 0.823534
\(630\) 0 0
\(631\) 21.8810 0.871068 0.435534 0.900172i \(-0.356560\pi\)
0.435534 + 0.900172i \(0.356560\pi\)
\(632\) −7.37800 −0.293481
\(633\) 0 0
\(634\) 17.2101 0.683499
\(635\) −54.9241 −2.17960
\(636\) 0 0
\(637\) −7.55775 −0.299449
\(638\) −1.74011 −0.0688917
\(639\) 0 0
\(640\) 3.02013 0.119381
\(641\) −31.3568 −1.23852 −0.619260 0.785186i \(-0.712567\pi\)
−0.619260 + 0.785186i \(0.712567\pi\)
\(642\) 0 0
\(643\) 13.8201 0.545012 0.272506 0.962154i \(-0.412148\pi\)
0.272506 + 0.962154i \(0.412148\pi\)
\(644\) 26.1663 1.03110
\(645\) 0 0
\(646\) 15.2799 0.601181
\(647\) −5.48150 −0.215500 −0.107750 0.994178i \(-0.534365\pi\)
−0.107750 + 0.994178i \(0.534365\pi\)
\(648\) 0 0
\(649\) 16.8067 0.659721
\(650\) −4.31200 −0.169130
\(651\) 0 0
\(652\) 10.0729 0.394487
\(653\) −0.281538 −0.0110174 −0.00550872 0.999985i \(-0.501753\pi\)
−0.00550872 + 0.999985i \(0.501753\pi\)
\(654\) 0 0
\(655\) −41.8454 −1.63504
\(656\) −5.58831 −0.218187
\(657\) 0 0
\(658\) 6.32215 0.246463
\(659\) −19.5223 −0.760481 −0.380241 0.924888i \(-0.624159\pi\)
−0.380241 + 0.924888i \(0.624159\pi\)
\(660\) 0 0
\(661\) 1.06418 0.0413919 0.0206959 0.999786i \(-0.493412\pi\)
0.0206959 + 0.999786i \(0.493412\pi\)
\(662\) −7.49987 −0.291491
\(663\) 0 0
\(664\) 2.11619 0.0821242
\(665\) 49.9948 1.93871
\(666\) 0 0
\(667\) −10.0404 −0.388766
\(668\) 8.97670 0.347319
\(669\) 0 0
\(670\) 23.2344 0.897623
\(671\) −12.1177 −0.467799
\(672\) 0 0
\(673\) 4.31488 0.166326 0.0831631 0.996536i \(-0.473498\pi\)
0.0831631 + 0.996536i \(0.473498\pi\)
\(674\) 2.60559 0.100364
\(675\) 0 0
\(676\) −11.9053 −0.457895
\(677\) −3.50610 −0.134750 −0.0673752 0.997728i \(-0.521462\pi\)
−0.0673752 + 0.997728i \(0.521462\pi\)
\(678\) 0 0
\(679\) −30.9814 −1.18896
\(680\) −10.5136 −0.403177
\(681\) 0 0
\(682\) 3.95031 0.151265
\(683\) −10.4697 −0.400610 −0.200305 0.979734i \(-0.564193\pi\)
−0.200305 + 0.979734i \(0.564193\pi\)
\(684\) 0 0
\(685\) −41.1139 −1.57088
\(686\) −0.842304 −0.0321593
\(687\) 0 0
\(688\) 9.61326 0.366502
\(689\) 8.25719 0.314574
\(690\) 0 0
\(691\) 38.5631 1.46701 0.733504 0.679685i \(-0.237883\pi\)
0.733504 + 0.679685i \(0.237883\pi\)
\(692\) −22.5083 −0.855637
\(693\) 0 0
\(694\) 6.30819 0.239456
\(695\) 33.3810 1.26621
\(696\) 0 0
\(697\) 19.4538 0.736865
\(698\) 7.33681 0.277702
\(699\) 0 0
\(700\) −15.5426 −0.587456
\(701\) −23.1419 −0.874056 −0.437028 0.899448i \(-0.643969\pi\)
−0.437028 + 0.899448i \(0.643969\pi\)
\(702\) 0 0
\(703\) 26.0424 0.982207
\(704\) −1.20245 −0.0453191
\(705\) 0 0
\(706\) −24.8373 −0.934763
\(707\) −6.92539 −0.260456
\(708\) 0 0
\(709\) 12.8373 0.482113 0.241057 0.970511i \(-0.422506\pi\)
0.241057 + 0.970511i \(0.422506\pi\)
\(710\) −8.06147 −0.302542
\(711\) 0 0
\(712\) 6.47710 0.242739
\(713\) 22.7932 0.853614
\(714\) 0 0
\(715\) 3.79969 0.142100
\(716\) 18.4992 0.691348
\(717\) 0 0
\(718\) 3.12865 0.116760
\(719\) −24.3201 −0.906987 −0.453494 0.891260i \(-0.649823\pi\)
−0.453494 + 0.891260i \(0.649823\pi\)
\(720\) 0 0
\(721\) −7.04581 −0.262400
\(722\) 0.266169 0.00990579
\(723\) 0 0
\(724\) 19.2505 0.715438
\(725\) 5.96394 0.221495
\(726\) 0 0
\(727\) 4.96463 0.184128 0.0920640 0.995753i \(-0.470654\pi\)
0.0920640 + 0.995753i \(0.470654\pi\)
\(728\) 3.94598 0.146248
\(729\) 0 0
\(730\) 10.5332 0.389852
\(731\) −33.4653 −1.23776
\(732\) 0 0
\(733\) 9.90002 0.365666 0.182833 0.983144i \(-0.441473\pi\)
0.182833 + 0.983144i \(0.441473\pi\)
\(734\) 10.0031 0.369219
\(735\) 0 0
\(736\) −6.93812 −0.255743
\(737\) −9.25067 −0.340753
\(738\) 0 0
\(739\) −2.33921 −0.0860493 −0.0430247 0.999074i \(-0.513699\pi\)
−0.0430247 + 0.999074i \(0.513699\pi\)
\(740\) −17.9188 −0.658708
\(741\) 0 0
\(742\) 29.7631 1.09264
\(743\) −28.5828 −1.04860 −0.524301 0.851533i \(-0.675673\pi\)
−0.524301 + 0.851533i \(0.675673\pi\)
\(744\) 0 0
\(745\) 39.7494 1.45630
\(746\) 18.1472 0.664418
\(747\) 0 0
\(748\) 4.18593 0.153053
\(749\) −2.03777 −0.0744585
\(750\) 0 0
\(751\) 2.12757 0.0776362 0.0388181 0.999246i \(-0.487641\pi\)
0.0388181 + 0.999246i \(0.487641\pi\)
\(752\) −1.67635 −0.0611301
\(753\) 0 0
\(754\) −1.51413 −0.0551415
\(755\) 31.7190 1.15437
\(756\) 0 0
\(757\) 22.6613 0.823638 0.411819 0.911266i \(-0.364894\pi\)
0.411819 + 0.911266i \(0.364894\pi\)
\(758\) −7.77655 −0.282457
\(759\) 0 0
\(760\) −13.2563 −0.480858
\(761\) 28.4967 1.03300 0.516502 0.856286i \(-0.327234\pi\)
0.516502 + 0.856286i \(0.327234\pi\)
\(762\) 0 0
\(763\) 8.39860 0.304050
\(764\) −13.2214 −0.478333
\(765\) 0 0
\(766\) −29.0604 −1.04999
\(767\) 14.6241 0.528046
\(768\) 0 0
\(769\) 44.0265 1.58764 0.793819 0.608154i \(-0.208090\pi\)
0.793819 + 0.608154i \(0.208090\pi\)
\(770\) 13.6960 0.493570
\(771\) 0 0
\(772\) 12.7010 0.457120
\(773\) 49.3490 1.77496 0.887481 0.460845i \(-0.152453\pi\)
0.887481 + 0.460845i \(0.152453\pi\)
\(774\) 0 0
\(775\) −13.5390 −0.486336
\(776\) 8.21487 0.294896
\(777\) 0 0
\(778\) 9.50596 0.340805
\(779\) 24.5289 0.878839
\(780\) 0 0
\(781\) 3.20964 0.114850
\(782\) 24.1527 0.863700
\(783\) 0 0
\(784\) 7.22334 0.257976
\(785\) 54.1371 1.93224
\(786\) 0 0
\(787\) 34.8197 1.24119 0.620594 0.784132i \(-0.286892\pi\)
0.620594 + 0.784132i \(0.286892\pi\)
\(788\) 6.80378 0.242375
\(789\) 0 0
\(790\) −22.2825 −0.792777
\(791\) −41.6399 −1.48055
\(792\) 0 0
\(793\) −10.5440 −0.374430
\(794\) 21.6620 0.768755
\(795\) 0 0
\(796\) −18.3936 −0.651944
\(797\) −3.93026 −0.139217 −0.0696086 0.997574i \(-0.522175\pi\)
−0.0696086 + 0.997574i \(0.522175\pi\)
\(798\) 0 0
\(799\) 5.83564 0.206450
\(800\) 4.12120 0.145706
\(801\) 0 0
\(802\) −20.1045 −0.709914
\(803\) −4.19375 −0.147994
\(804\) 0 0
\(805\) 79.0258 2.78529
\(806\) 3.43731 0.121074
\(807\) 0 0
\(808\) 1.83630 0.0646008
\(809\) 45.9514 1.61556 0.807782 0.589481i \(-0.200668\pi\)
0.807782 + 0.589481i \(0.200668\pi\)
\(810\) 0 0
\(811\) 27.2715 0.957630 0.478815 0.877916i \(-0.341066\pi\)
0.478815 + 0.877916i \(0.341066\pi\)
\(812\) −5.45771 −0.191528
\(813\) 0 0
\(814\) 7.13429 0.250057
\(815\) 30.4216 1.06562
\(816\) 0 0
\(817\) −42.1957 −1.47624
\(818\) −36.8815 −1.28953
\(819\) 0 0
\(820\) −16.8774 −0.589386
\(821\) 32.9435 1.14974 0.574868 0.818246i \(-0.305053\pi\)
0.574868 + 0.818246i \(0.305053\pi\)
\(822\) 0 0
\(823\) 37.5067 1.30740 0.653700 0.756754i \(-0.273216\pi\)
0.653700 + 0.756754i \(0.273216\pi\)
\(824\) 1.86823 0.0650828
\(825\) 0 0
\(826\) 52.7128 1.83411
\(827\) −5.28182 −0.183667 −0.0918334 0.995774i \(-0.529273\pi\)
−0.0918334 + 0.995774i \(0.529273\pi\)
\(828\) 0 0
\(829\) −21.4341 −0.744437 −0.372218 0.928145i \(-0.621403\pi\)
−0.372218 + 0.928145i \(0.621403\pi\)
\(830\) 6.39118 0.221841
\(831\) 0 0
\(832\) −1.04630 −0.0362738
\(833\) −25.1456 −0.871244
\(834\) 0 0
\(835\) 27.1108 0.938209
\(836\) 5.27795 0.182542
\(837\) 0 0
\(838\) −33.9553 −1.17297
\(839\) −9.17185 −0.316647 −0.158324 0.987387i \(-0.550609\pi\)
−0.158324 + 0.987387i \(0.550609\pi\)
\(840\) 0 0
\(841\) −26.9058 −0.927786
\(842\) −36.8261 −1.26911
\(843\) 0 0
\(844\) −23.9426 −0.824137
\(845\) −35.9555 −1.23691
\(846\) 0 0
\(847\) 36.0322 1.23808
\(848\) −7.89183 −0.271006
\(849\) 0 0
\(850\) −14.3466 −0.492083
\(851\) 41.1647 1.41111
\(852\) 0 0
\(853\) −6.81667 −0.233398 −0.116699 0.993167i \(-0.537231\pi\)
−0.116699 + 0.993167i \(0.537231\pi\)
\(854\) −38.0061 −1.30054
\(855\) 0 0
\(856\) 0.540324 0.0184679
\(857\) 7.61268 0.260044 0.130022 0.991511i \(-0.458495\pi\)
0.130022 + 0.991511i \(0.458495\pi\)
\(858\) 0 0
\(859\) 18.6435 0.636109 0.318055 0.948072i \(-0.396970\pi\)
0.318055 + 0.948072i \(0.396970\pi\)
\(860\) 29.0333 0.990028
\(861\) 0 0
\(862\) 1.58581 0.0540131
\(863\) 39.6210 1.34872 0.674358 0.738405i \(-0.264421\pi\)
0.674358 + 0.738405i \(0.264421\pi\)
\(864\) 0 0
\(865\) −67.9780 −2.31132
\(866\) −2.99805 −0.101878
\(867\) 0 0
\(868\) 12.3898 0.420538
\(869\) 8.87169 0.300951
\(870\) 0 0
\(871\) −8.04934 −0.272741
\(872\) −2.22693 −0.0754133
\(873\) 0 0
\(874\) 30.4537 1.03011
\(875\) 10.0096 0.338386
\(876\) 0 0
\(877\) 43.4068 1.46574 0.732871 0.680367i \(-0.238180\pi\)
0.732871 + 0.680367i \(0.238180\pi\)
\(878\) 17.2951 0.583680
\(879\) 0 0
\(880\) −3.63156 −0.122420
\(881\) −16.6130 −0.559706 −0.279853 0.960043i \(-0.590286\pi\)
−0.279853 + 0.960043i \(0.590286\pi\)
\(882\) 0 0
\(883\) 41.3726 1.39230 0.696149 0.717897i \(-0.254895\pi\)
0.696149 + 0.717897i \(0.254895\pi\)
\(884\) 3.64232 0.122505
\(885\) 0 0
\(886\) −22.8314 −0.767037
\(887\) −14.7151 −0.494085 −0.247042 0.969005i \(-0.579459\pi\)
−0.247042 + 0.969005i \(0.579459\pi\)
\(888\) 0 0
\(889\) 68.5864 2.30031
\(890\) 19.5617 0.655709
\(891\) 0 0
\(892\) −1.00000 −0.0334825
\(893\) 7.35803 0.246227
\(894\) 0 0
\(895\) 55.8701 1.86753
\(896\) −3.77138 −0.125993
\(897\) 0 0
\(898\) −28.4260 −0.948587
\(899\) −4.75416 −0.158560
\(900\) 0 0
\(901\) 27.4727 0.915249
\(902\) 6.71967 0.223741
\(903\) 0 0
\(904\) 11.0410 0.367219
\(905\) 58.1390 1.93260
\(906\) 0 0
\(907\) −21.9360 −0.728374 −0.364187 0.931326i \(-0.618653\pi\)
−0.364187 + 0.931326i \(0.618653\pi\)
\(908\) −18.9100 −0.627549
\(909\) 0 0
\(910\) 11.9174 0.395058
\(911\) 30.5867 1.01338 0.506691 0.862128i \(-0.330869\pi\)
0.506691 + 0.862128i \(0.330869\pi\)
\(912\) 0 0
\(913\) −2.54462 −0.0842146
\(914\) −4.12755 −0.136527
\(915\) 0 0
\(916\) −9.36817 −0.309533
\(917\) 52.2544 1.72559
\(918\) 0 0
\(919\) −45.1531 −1.48946 −0.744732 0.667363i \(-0.767423\pi\)
−0.744732 + 0.667363i \(0.767423\pi\)
\(920\) −20.9541 −0.690835
\(921\) 0 0
\(922\) 23.7885 0.783432
\(923\) 2.79282 0.0919268
\(924\) 0 0
\(925\) −24.4516 −0.803962
\(926\) −5.25289 −0.172621
\(927\) 0 0
\(928\) 1.44714 0.0475046
\(929\) 2.38827 0.0783565 0.0391782 0.999232i \(-0.487526\pi\)
0.0391782 + 0.999232i \(0.487526\pi\)
\(930\) 0 0
\(931\) −31.7056 −1.03911
\(932\) 22.4238 0.734517
\(933\) 0 0
\(934\) −28.3361 −0.927186
\(935\) 12.6421 0.413439
\(936\) 0 0
\(937\) −31.0450 −1.01419 −0.507097 0.861889i \(-0.669282\pi\)
−0.507097 + 0.861889i \(0.669282\pi\)
\(938\) −29.0139 −0.947338
\(939\) 0 0
\(940\) −5.06279 −0.165130
\(941\) −15.1571 −0.494106 −0.247053 0.969002i \(-0.579462\pi\)
−0.247053 + 0.969002i \(0.579462\pi\)
\(942\) 0 0
\(943\) 38.7724 1.26260
\(944\) −13.9770 −0.454914
\(945\) 0 0
\(946\) −11.5595 −0.375831
\(947\) 40.5902 1.31900 0.659502 0.751703i \(-0.270767\pi\)
0.659502 + 0.751703i \(0.270767\pi\)
\(948\) 0 0
\(949\) −3.64913 −0.118456
\(950\) −18.0893 −0.586894
\(951\) 0 0
\(952\) 13.1288 0.425507
\(953\) 9.71509 0.314703 0.157351 0.987543i \(-0.449705\pi\)
0.157351 + 0.987543i \(0.449705\pi\)
\(954\) 0 0
\(955\) −39.9303 −1.29212
\(956\) −12.7281 −0.411657
\(957\) 0 0
\(958\) −12.1657 −0.393056
\(959\) 51.3408 1.65788
\(960\) 0 0
\(961\) −20.2074 −0.651850
\(962\) 6.20780 0.200147
\(963\) 0 0
\(964\) −14.7087 −0.473736
\(965\) 38.3588 1.23481
\(966\) 0 0
\(967\) 23.7630 0.764166 0.382083 0.924128i \(-0.375207\pi\)
0.382083 + 0.924128i \(0.375207\pi\)
\(968\) −9.55411 −0.307081
\(969\) 0 0
\(970\) 24.8100 0.796600
\(971\) −9.20942 −0.295544 −0.147772 0.989021i \(-0.547210\pi\)
−0.147772 + 0.989021i \(0.547210\pi\)
\(972\) 0 0
\(973\) −41.6844 −1.33634
\(974\) −18.2502 −0.584773
\(975\) 0 0
\(976\) 10.0775 0.322573
\(977\) −22.2339 −0.711325 −0.355662 0.934615i \(-0.615745\pi\)
−0.355662 + 0.934615i \(0.615745\pi\)
\(978\) 0 0
\(979\) −7.78840 −0.248918
\(980\) 21.8154 0.696869
\(981\) 0 0
\(982\) 5.52096 0.176181
\(983\) −22.8407 −0.728506 −0.364253 0.931300i \(-0.618676\pi\)
−0.364253 + 0.931300i \(0.618676\pi\)
\(984\) 0 0
\(985\) 20.5483 0.654724
\(986\) −5.03771 −0.160433
\(987\) 0 0
\(988\) 4.59253 0.146108
\(989\) −66.6980 −2.12087
\(990\) 0 0
\(991\) 30.1784 0.958647 0.479323 0.877638i \(-0.340882\pi\)
0.479323 + 0.877638i \(0.340882\pi\)
\(992\) −3.28522 −0.104306
\(993\) 0 0
\(994\) 10.0667 0.319298
\(995\) −55.5511 −1.76109
\(996\) 0 0
\(997\) 15.3755 0.486947 0.243473 0.969908i \(-0.421713\pi\)
0.243473 + 0.969908i \(0.421713\pi\)
\(998\) 32.2553 1.02102
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 4014.2.a.r.1.5 5
3.2 odd 2 1338.2.a.h.1.1 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1338.2.a.h.1.1 5 3.2 odd 2
4014.2.a.r.1.5 5 1.1 even 1 trivial